where we used the Coarea Formula (see [1]}) and the fact that the \((d-1)\) -dimensional Hausdorff measure \({\mathcal {H}}^{d-1}(\Psi ({\mathcal {S}}^{\tau }, \tau , \tau ))\)  is bounded by a constant \(K>0\)  since, by hypotheses, \(\Psi \)  is Lipschitz continuous and \({\mathcal {S}}^{\tau }\)  is compact. Then, the map \(s\longmapsto {\mathcal {L}}^d({\mathcal {B}}(s))\)  is Lipschitz continuous and the thesis follows.
